Find the resultant of two vectors of 3 unit and 4 units acting at a point O at angle of 45 with each other:solve using scale drawing method and analytical method:4cm
To find the resultant of two vectors using the scale drawing method, follow these steps:
1. Draw a line segment to represent the first vector, measuring 3 units in length. Label one end of the line segment as point O.
2. From the endpoint of the first vector, draw a second line segment at an angle of 45 degrees to represent the second vector, measuring 4 units in length.
3. Complete a parallelogram by drawing lines parallel to the vectors from their respective endpoints.
4. Measure the length of the diagonal of the parallelogram. This represents the magnitude of the resultant vector.
5. Use a protractor to measure the angle between the resultant vector and the first vector. This represents the direction of the resultant vector.
To solve using the analytical method:
Given that the first vector is 3 units in magnitude and the second vector is 4 units in magnitude, we have vector A = 3 units and vector B = 4 units.
By using the Law of Cosines, we can find the magnitude of the resultant vector R:
R^2 = A^2 + B^2 - 2AB*cos(angle between A and B)
Substituting the given values, we have:
R^2 = 3^2 + 4^2 - 2(3)(4)*cos(45°)
R^2 = 9 + 16 - 24*cos(45°)
R^2 = 25 - 24*(sqrt(2)/2)
R^2 = 25 - 12*sqrt(2)
R^2 ≈ 3.8988
Taking the square root of both sides, we find that the magnitude of the resultant vector is approximately 1.9744.
To find the direction of the resultant vector, we can use the Law of Sines:
sin(angle between A and B) / R = sin(angle between A and resultant) / A
Substituting the given values, we have:
sin(angle between A and resultant) / 3 = sin(45°) / 1.9744
Simplifying, we have:
sin(angle between A and resultant) ≈ 0.4795
Taking the inverse sine of both sides, we find that the angle between vector A and the resultant vector is approximately 28.14°.
Therefore, the resultant vector has a magnitude of approximately 1.9744 units and is inclined at an angle of approximately 28.14° with vector A.
To find the resultant of two vectors using the scale drawing method, we will begin by drawing a diagram.
1. Start by drawing a horizontal line segment of 3 units (representing the first vector) from point O.
2. From the endpoint of the first vector, draw a second vector of 4 units at an angle of 45 degrees clockwise from the first vector.
3. Draw a line connecting the starting point O with the endpoint of the second vector.
4. Measure the magnitude of the resultant vector on the drawing.
To find the resultant using the analytical method:
Let's denote vector A as 3 units and vector B as 4 units, both acting at angle 45 degrees with each other.
1. Resolve both vectors into x and y components.
- Vector A: x-component = A * cos(45) = 3 * cos(45) = 3 * √2 / 2 = 3√2 / 2
y-component = A * sin(45) = 3 * sin(45) = 3 * √2 / 2 = 3√2 / 2
- Vector B: x-component = B * cos(45) = 4 * cos(45) = 4 * √2 / 2 = 4√2 / 2
y-component = B * sin(45) = 4 * sin(45) = 4 * √2 / 2 = 4√2 / 2
2. Add the x and y components separately.
- x-component resultant = (3√2 / 2) + (4√2 / 2) = 7√2 / 2
- y-component resultant = (3√2 / 2) + (4√2 / 2) = 7√2 / 2
3. Calculate the magnitude of the resultant vector using Pythagorean theorem.
- Magnitude of the resultant = √[(x-component resultant)^2 + (y-component resultant)^2]
= √[(7√2 / 2)^2 + (7√2 / 2)^2]
= √[(49/2) + (49/2)]
= √[(98/2)]
= √(49)
= 7
So, the magnitude of the resultant vector is 7 units.
Note: The scale drawing method provides an approximate solution, while the analytical method gives a more accurate answer.
To find the resultant of two vectors using the scale drawing method, you can follow these steps:
1. Draw a vector representing the first vector (3 units) starting from point O. Choose a convenient scale, such as 1 cm:1 unit.
2. Draw a vector representing the second vector (4 units) starting from the end of the first vector. The angle between the two vectors is 45 degrees.
3. Complete the triangle by drawing a line from the end of the second vector back to point O.
4. Measure the length of the side opposite to the angle of the triangle (the resultant) using a ruler. Convert the measurement from cm to the original unit scale (1 cm: 1 unit).
To solve the problem analytically, we can use the magnitude and components of the vectors:
Let the first vector be v1 = 3 units, and the second vector be v2 = 4 units at an angle of 45 degrees.
To find the resultant analytically, we can break down the vectors into their x and y components and then sum them in each direction.
For v1:
vx1 = v1 * cos(angle) = 3 * cos(45) ≈ 2.12 units
vy1 = v1 * sin(angle) = 3 * sin(45) ≈ 2.12 units
For v2:
vx2 = v2 * cos(angle) = 4 * cos(45) ≈ 2.83 units
vy2 = v2 * sin(angle) = 4 * sin(45) ≈ 2.83 units
The resultant vector, vR, is given by the sum of the x-components and the sum of the y-components:
vxR = vx1 + vx2 = 2.12 + 2.83 ≈ 4.95 units
vyR = vy1 + vy2 = 2.12 + 2.83 ≈ 4.95 units
The magnitude of the resultant vector can be found using Pythagoras' theorem:
vR = sqrt(vxR^2 + vyR^2) ≈ sqrt(4.95^2 + 4.95^2) ≈ 7.0 units
Therefore, the resultant of the two vectors is approximately 7 units.